In an equilateral triangle ABC, the radius of the inscribed circle is 3 cm. Find the radius of the circumscribed circle.

An equilateral (regular) triangle is a triangle in which all sides and angles are equal.
1. The radius of a circle inscribed in an equilateral triangle is found by the formula:
r = a√3 / 6,
where a is the length of the side of the triangle.
Substitute the data for the value condition and find the length of the side of the triangle.
a√3 / 6 = 3;
a√3 = 18 (proportional);
a = 18 / √3 (get rid of irrationality in the denominator by multiplying the fraction by √3);
a = 18√3 / 3 (we will reduce the fraction);
a = 6√3 cm.
2. The radius of a circle circumscribed about a regular triangle is found by the formula:
R = a√3 / 3.
Substitute the known value of a and find the length of the radius of the circumscribed circle:
R = 6√3 * √3 / 3 = 6 * 3/3 = 6 (cm).
Answer: R = 6 cm.